12 research outputs found
Transversals and colorings of simplicial spheres
Motivated from the surrounding property of a point set in
introduced by Holmsen, Pach and Tverberg, we consider the transversal number
and chromatic number of a simplicial sphere. As an attempt to give a lower
bound for the maximum transversal ratio of simplicial -spheres, we provide
two infinite constructions. The first construction gives infintely many
-dimensional simplicial polytopes with the transversal ratio exactly
for every . In the case of , this meets the
previously well-known upper bound tightly. The second gives infinitely
many simplicial 3-spheres with the transversal ratio greater than . This
was unexpected from what was previously known about the surrounding property.
Moreover, we show that, for , the facet hypergraph
of a -dimensional simplicial sphere
has the chromatic number , where is the number of vertices of . This
slightly improves the upper bound previously obtained by Heise, Panagiotou,
Pikhurko, and Taraz.Comment: 22 pages, 2 figure
Realization spaces of arrangements of convex bodies
We introduce combinatorial types of arrangements of convex bodies, extending
order types of point sets to arrangements of convex bodies, and study their
realization spaces. Our main results witness a trade-off between the
combinatorial complexity of the bodies and the topological complexity of their
realization space. First, we show that every combinatorial type is realizable
and its realization space is contractible under mild assumptions. Second, we
prove a universality theorem that says the restriction of the realization space
to arrangements polygons with a bounded number of vertices can have the
homotopy type of any primary semialgebraic set